OrdinaryDiffEqTaylorSeries

Taylor series methods for ordinary differential equations using automatic differentiation. These methods achieve very high-order accuracy by computing Taylor expansions of the solution using automatic differentiation techniques through TaylorDiff.jl.

Key Properties

Taylor series methods provide:

• Very high-order accuracy with arbitrary order capability
• Automatic differentiation for derivative computation
• Step size control through Taylor series truncation
• Natural error estimation from higher-order terms
• Excellent accuracy for smooth problems
• Single-step methods without requiring history

When to Use Taylor Series Methods

These methods are recommended for:

• Ultra-high precision problems where maximum accuracy is needed
• Smooth problems with well-behaved derivatives
• Scientific computing requiring very low error tolerances
• Problems with expensive function evaluations where high-order methods reduce total steps

Mathematical Background

Taylor series methods compute the solution using Taylor expansions: u(t + h) = u(t) + h*u'(t) + h²/2!*u''(t) + h³/3!*u'''(t) + ...

The derivatives are computed automatically using automatic differentiation, allowing arbitrary-order methods without manual derivative computation.

Solver Selection Guide

Available Methods

• ExplicitTaylor2: Second-order Taylor series method for moderate accuracy
• ExplicitTaylor: Arbitrary-order Taylor series method (specify order with order = Val{p}())

Usage considerations

• Smooth problems only: Methods assume the function has many continuous derivatives
• Computational cost: Higher orders require more automatic differentiation computations
• Memory requirements: Higher orders store more derivative information

Performance Guidelines

When Taylor series methods excel

• Very smooth problems where high-order derivatives exist and are well-behaved
• High precision requirements beyond standard double precision
• Long-time integration where accumulated error matters
• Problems where function evaluations dominate computational cost

Problem characteristics

• Polynomial and analytic functions work extremely well
• Smooth ODEs from physics simulations
• Problems requiring very low tolerances (< 1e-12)

Limitations and Considerations

Method limitations

• Requires smooth functions - non-smooth problems may cause issues
• Memory overhead for storing multiple derivatives
• Limited to problems where high-order derivatives are meaningful
• Automatic differentiation compatibility - requires functions compatible with TaylorDiff.jl and Symbolics.jl tracing
• Long compile times due to automatic differentiation and symbolic processing overhead

When to consider alternatives

• Non-smooth problems: Use adaptive Runge-Kutta methods instead
• Stiff problems: Taylor methods are explicit and may be inefficient
• Large systems: Automatic differentiation cost may become prohibitive
• Standard accuracy needs: Lower-order methods are often sufficient

Alternative Approaches

Consider these alternatives:

• High-order Runge-Kutta methods (Feagin, Verner) for good accuracy with less overhead
• Extrapolation methods for high accuracy with standard function evaluations
• Adaptive methods for problems with varying smoothness
• Implicit methods for stiff problems requiring high accuracy

Installation and Usage

Taylor series methods require explicit installation of the specialized library:

using Pkg
Pkg.add("OrdinaryDiffEqTaylorSeries")

Then use the methods with:

using OrdinaryDiffEqTaylorSeries

# Example: Second-order Taylor method
function f(u, p, t)
    σ, ρ, β = p
    du1 = σ * (u[2] - u[1])
    du2 = u[1] * (ρ - u[3]) - u[2]
    du3 = u[1] * u[2] - β * u[3]
    [du1, du2, du3]
end

u0 = [1.0, 0.0, 0.0]
tspan = (0.0, 10.0)
p = [10.0, 28.0, 8/3]
prob = ODEProblem(f, u0, tspan, p)

# Second-order Taylor method
sol = solve(prob, ExplicitTaylor2())

# Arbitrary-order Taylor method (e.g., 8th order)
sol = solve(prob, ExplicitTaylor(order = Val{8}()))

Full list of solvers